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A grandfather clock is found to gain 39.0 s every 24 hours. If the current length of the pendulum is 0.660 m, how much should the length be
Question
A grandfather clock is found to gain 39.0 s every 24 hours. If the current length of the pendulum is 0.660 m, how much should the length be changed (in m) to make the clock accurate? Let a positive value correspond to an increase in length, and a negative value correspond to a decrease.
ΔL=________
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Physics
4 years
2021-07-28T11:15:00+00:00
2021-07-28T11:15:00+00:00 1 Answers
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Answer:
Explanation:
The time period (T) of a simple pendulum depends on the length (l) of the pendulum and acceleration due to gravity (g). The time period of a simple pendulum can be calculated as follows
T = 2π√(l/g)
The number of hours in a day is 24 h.
Convert 24 h to second as follows,
1 hr = 60mins
1 min = 60secs
1 hr = 60 × 60 = 3600secs.
24hr = 24 × 3600 = 86,400sec
The number of cycle’s clock made per day is,
86,400 / 2 = 43,200 sec
If the clock runs, slow of 39 s, then there is 19.5 cycles is reduces per day. So,
(43200 – 19.5) = 43,180.5 seconds
Then, The new time period would be reduces by 43,180.5 / 43200 to the old time period
To = 43,180.5/43200 Tn
To = 0.99955 Tn.
To is old period
Tn is new period.
Given that, initial length is.
Lo = 0.66m
Relating the old period to the new period
To = 2π√(Lo / g). Equation 1
Tn = 2π√(Ln / g). Equation 2
We know that
To = 0.99955 Tn
To / Tn = 0.99955
Divide equation 1 with 2
To/Tn = 2π√(Lo / g) / 2π√(Ln / g)
To / Tn = √(Lo / Ln)
0.99955 = √(Lo / Ln)
0.99955² = Lo / Ln
0.9991 = 0.66 / Ln
Ln = 0.66 / 0.9991
Ln = 0.6606m
The difference in the lengths is,
∆L = Ln — Lo
∆L = 0.6606—0.66
∆L = 5.96 × 10^-4m
∆L = 0.596 mm
Therefore, the new length of the pendulum is 0.6606m that is the length of the pendulum should be reduced by 0.596mm.